$ D = \left[\begin{array}{rrr}4 & -1 & 5 \\ 0 & 0 & 2\end{array}\right]$ $ F = \left[\begin{array}{rr}3 & 5 \\ 2 & 1 \\ 1 & 2\end{array}\right]$ What is $ D F$ ?
Explanation: Because $ D$ has dimensions $(2\times3)$ and $ F$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ D F = \left[\begin{array}{rrr}{4} & {-1} & {5} \\ {0} & {0} & {2}\end{array}\right] \left[\begin{array}{rr}{3} & \color{#DF0030}{5} \\ {2} & \color{#DF0030}{1} \\ {1} & \color{#DF0030}{2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{4}\cdot{3}+{-1}\cdot{2}+{5}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{3}+{-1}\cdot{2}+{5}\cdot{1} & ? \\ {0}\cdot{3}+{0}\cdot{2}+{2}\cdot{1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{3}+{-1}\cdot{2}+{5}\cdot{1} & {4}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{1}+{5}\cdot\color{#DF0030}{2} \\ {0}\cdot{3}+{0}\cdot{2}+{2}\cdot{1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{4}\cdot{3}+{-1}\cdot{2}+{5}\cdot{1} & {4}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{1}+{5}\cdot\color{#DF0030}{2} \\ {0}\cdot{3}+{0}\cdot{2}+{2}\cdot{1} & {0}\cdot\color{#DF0030}{5}+{0}\cdot\color{#DF0030}{1}+{2}\cdot\color{#DF0030}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}15 & 29 \\ 2 & 4\end{array}\right] $